If you are using the ArcGIS for JavaScript API, adding the layer as an ArcGIS Tilled Map Service Layer allows you to resample tiles when you zoom in past the last level. You will find these options in the constructor as resampling and resamplingTolerance.

Given that re-sampling is disabled by default, I'm suspecting that the web map is just using the defaults. I couldn't find anywhere to override this setting in web maps.